Chapter 1, Problem 157QP

A bicyclist is traveling at 6.7 meters per second. Which of the mathematical operation would correctly determine the speed in miles per hour?

$\begin{array}{l}\text{A}\text{.}\frac{\text{6}\text{.7}\text{m}}{\text{s}}\times \frac{1000\text{km}}{1\text{m}}\times \frac{0.621\text{mi}}{1\text{km}}\times \frac{60\text{s}}{1\min }\times \frac{60\text{min}}{1\text{hr}}\\ \text{B}\text{.}\frac{\text{6}\text{.7}\text{m}}{\text{s}}\times \frac{\text{1}\text{km}}{1000\text{m}}\times \frac{1\text{km}}{0.621\text{mi}}\times \frac{60\text{s}}{1\min }\times \frac{60\min }{1\text{hr}}\\ \text{C}\text{.}\frac{\text{6}\text{.7}\text{m}}{\text{s}}\times \frac{\text{1}\text{km}}{1000\text{m}}\times \frac{0.621\text{mi}}{1\text{km}}\times \frac{60\text{s}}{1\min }\times \frac{60\min }{1\text{hr}}\\ \text{D}\text{.}\frac{6.7\text{m}}{\text{s}}\times \frac{1\text{km}}{1000\text{m}}\times \frac{0.621\text{mi}}{1\text{km}}\times \frac{1\min }{60\text{s}}\times \frac{1\text{hr}}{60\min }\\ \text{E}\text{.}\frac{\text{6}\text{.7m}}{\text{s}}\times \frac{1000\text{m}}{1\text{km}}\times \frac{\text{0}\text{.621}\text{mi}}{\text{1}\text{km}}\times \frac{1\min }{60\text{s}}\times \frac{\text{1}\text{hr}}{60\min }\end{array}$

Explain what is wrong with the incorrect responses.

Interpretation Introduction

Interpretation:

Whether the given mathematical operation gives the value of speed in miles per hour is correct or not is to be determined.

Explanation of Solution

The distance traveled by an object per unit time is known as speed. The SI unit of speed is meter per second $\left(\text{m}/\text{s}\right)$ . It can also be measured in centimeters per second $\left(\text{cm}/\text{s}\right)$ , kilometer per hour $\left(\text{km/h}\right)$ , and miles per hour $\left(\text{mi}/\text{h}\right)$ .

The conversion factor from $\text{m}$ to $\text{km}$ is expressed as follows:

$\begin{array}{c}1\text{ m}=0.001\text{ km}\\ 1=\frac{1\text{ km}}{1000\text{ m}}\end{array}$

The conversion factor from $\left(\text{m}/\text{s}\right)$ to $\left(\text{mi}/\text{h}\right)$ is expressed as follows:

$\begin{array}{c}\text{1 m/s}=\text{2}\text{.236 mi/hr}\\ \text{1=}\frac{\text{2}\text{.236 mi/hr}}{\text{1 m/s}}\end{array}$

In the given mathematical operation $\frac{\text{6}\text{.7 m}}{\text{s}}\text{×}\frac{\text{1000 km}}{\text{1 m}}\text{×}\frac{\text{0}\text{.621mi}}{\text{1 km}}\text{×}\frac{\text{60 s}}{\text{1 min}}\text{×}\frac{\text{60 min}}{\text{1 hr}}$ , the conversion factor for meters to kilometers is incorrect. It should be $\frac{\text{1 km}}{\text{1000 m}}$ . Therefore, the mathematical operation $\frac{\text{6}\text{.7 m}}{\text{s}}\text{×}\frac{\text{1 km}}{\text{1000 m}}\text{×}\frac{\text{0}\text{.621mi}}{\text{1 km}}\text{×}\frac{\text{60 s}}{\text{1 min}}\text{×}\frac{\text{60 min}}{\text{1 hr}}$ gives the correct values of speed in $\left(\text{mi}/\text{h}\right)$ .

Interpretation Introduction

Interpretation:

Whether the given mathematical operation gives the value of speed in miles per hour is correct or not is to be determined.

Explanation of Solution

The conversion factor from $\text{km}$ to $\text{miles}$ is expressed as follows:

$\begin{array}{c}1\text{ km}=0.621\text{ miles}\\ 1=\frac{0.621\text{ miles}}{1\text{ km}}\end{array}$

In the given mathematical operation $\frac{\text{6}\text{.7 m}}{\text{s}}\text{×}\frac{\text{1000 km}}{\text{1 m}}\text{×}\frac{\text{1 km}}{\text{0}\text{.621 mi}}\text{×}\frac{\text{60 s}}{\text{1 min}}\text{×}\frac{\text{60 min}}{\text{1 hr}}$ , the conversion factor for kilometers to miles is incorrect. It should be $\frac{0.621\text{miles}}{1\text{ km}}$ . Therefore, the correct mathematical operation $\frac{\text{6}\text{.7 m}}{\text{s}}\text{×}\frac{\text{1 km}}{\text{1000 m}}\text{×}\frac{\text{0}\text{.621mi}}{\text{1 km}}\text{×}\frac{\text{60 s}}{\text{1 min}}\text{×}\frac{\text{60 min}}{\text{1 hr}}$ gives the values of speed in $\left(\text{mi}/\text{h}\right)$ .

Interpretation Introduction

Interpretation:

Whether the given mathematical operation gives the value of speed in miles per hour is correct or not is to be determined.

Explanation of Solution

The given mathematical operation $\frac{\text{6}\text{.7 m}}{\text{s}}\text{×}\frac{\text{1 km}}{\text{1000 m}}\text{×}\frac{\text{0}\text{.621 mi}}{\text{1 km}}\text{×}\frac{\text{60 s}}{\text{1 min}}\text{×}\frac{\text{60 min}}{\text{1 hr}}$ is appropriate to find the speed in miles per hour.

Interpretation Introduction

Interpretation:

Whether the given mathematical operation gives the value of speed in miles per hour is correct or not is to be determined.

Explanation of Solution

The conversion factor from $\text{minute}$ to $\text{second}$ is expressed as follows:

$\begin{array}{c}\text{1 min}=\text{60 s}\\ 1=\frac{\text{60 s}}{\text{1 min}}\end{array}$

The conversion factor from $\text{hr}$ to $\min$ is expressed as follows:

$\begin{array}{l}1\text{ hr}=\text{60 min}\\ 1=\frac{\text{60 min}}{\text{1 hr}}\end{array}$

In the given mathematical operation $\frac{\text{6}\text{.7 m}}{\text{s}}\text{×}\frac{\text{1 km}}{\text{1000 m}}\text{×}\frac{\text{0}\text{.621 mi}}{\text{1 km}}\text{×}\frac{\text{1 min}}{\text{60 s}}\text{×}\frac{\text{1 hr}}{60\min }$ , the conversion factors in the fourth and fifth steps are incorrect. It should be $\frac{\text{60 s}}{\text{1 min}}$ and $\frac{\text{60 min}}{\text{1 hr}}$ , respectively.

Therefore, the correct mathematical operation $\frac{\text{6}\text{.7 m}}{\text{s}}\text{×}\frac{\text{1 km}}{\text{1000 m}}\text{×}\frac{\text{0}\text{.621mi}}{\text{1 km}}\text{×}\frac{\text{60 s}}{\text{1 min}}\text{×}\frac{\text{60 min}}{\text{1 hr}}$ gives the values of speed in $\left(\text{mi}/\text{h}\right)$ .

Interpretation Introduction

Interpretation:

Whether the given mathematical operation gives the value of speed in miles per hour is correct or not is to be determined.

Explanation of Solution

The conversion factor from $\text{hr}$ to $\min$ is expressed as follows:

$\begin{array}{l}1\text{ hr}=\text{60 min}\\ 1=\frac{\text{60 min}}{\text{1 hr}}\end{array}$

The conversion factor from $\text{m}$ to $\text{km}$ is expressed as follows:

$\begin{array}{c}1\text{ m}=0.001\text{ km}\\ 1=\frac{1\text{ km}}{1000\text{ m}}\end{array}$

The conversion factor from $\text{minute}$ to $\text{second}$ is expressed as follows:

$\begin{array}{c}\text{1 min}=\text{60 s}\\ 1=\frac{\text{60 s}}{\text{1 min}}\end{array}$

In the given mathematical operation $\frac{\text{6}\text{.7 m}}{\text{s}}\text{×}\frac{\text{1000 m}}{\text{1 km}}\text{×}\frac{\text{0}\text{.621 mi}}{\text{1 km}}\text{×}\frac{\text{1 min}}{\text{60 s}}\text{×}\frac{\text{1 hr}}{60\min }$ , the conversion factors in the second and last two steps are incorrect. It should be $\frac{1\text{ km}}{1000\text{ m}}$ , $\frac{\text{60 s}}{\text{1 min}}$ , and $\frac{\text{60 min}}{\text{1 hr}}$ , respectively.

Therefore, the correct mathematical operation $\frac{\text{6}\text{.7 m}}{\text{s}}\text{×}\frac{\text{1 km}}{\text{1000 m}}\text{×}\frac{\text{0}\text{.621mi}}{\text{1 km}}\text{×}\frac{\text{60 s}}{\text{1 min}}\text{×}\frac{\text{60 min}}{\text{1 hr}}$ gives the values of speed in $\left(\text{mi}/\text{h}\right)$ .